Third Cumulant for Multivariate Aggregate Claim Models

Nicola Loperfido, Stepan Mazur, Krzysztof Podgorski

Research output: Working paper/PreprintWorking paper

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Abstract

The third moment cumulant for the aggregated multivariate claims is considered. A formula is presented for the general case when the aggregating variable is independent of the multivariate claims. It is discussed how this result can be used to obtain a formula for the third cumulant for a classical model of multivariate claims. Two important special cases are considered. In the rst one, multivariate skewed normal claims are considered and aggregated by a Poisson variable. The second case is dealing with multivariate asymmetric generalized Laplace and aggregation is made by a negative binomial variable. Due to the invariance property the latter case can be derived directly leading to the identity involving the cumulant of the claims and the aggregated claims. There is a well established relation between asymmetric Laplace motion and negative binomial process that corresponds to the invariance principle of the aggregating claims for the generalized asymmetric Laplace distribution. We explore this relation and provide multivariate continuous time version of the results.
Original languageEnglish
PublisherDepartment of Statistics, Lund university
Number of pages30
Publication statusPublished - 2015

Publication series

NameWorking Papers in Statistics
No.13

Subject classification (UKÄ)

  • Probability Theory and Statistics

Free keywords

  • Third cumulant
  • multivariate aggregate claim
  • skew-normal
  • Laplace motion

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